How many significant digits are correct in my answer?

To find how many significant digits are correct in my answer in a numerical method that gives iterative values, one finds the absolute relative approximate percentage error defined as
|(Current approximation-Previous approximation)/Current approximation|*100
If the absolute relative approximate percentage error is less than or equal to 0.5*10^(2-m), then m significant digits are at least correct in the answer.
For example, if you want

at least 1 signficant digit to be correct in your answer, your absolute relative approximate error should be less than or equal to 5%

at least 2 signficant digit to be correct in your answer, your absolute relative approximate error should be less than or equal to 0.5%

at least 3 signficant digit to be correct in your answer, your absolute relative approximate error should be less than or equal to 0.05%
and so on.

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Author: Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.
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